Precision Limits of Tissue Microstructure Characterization by Magnetic Resonance Imaging

Characterization of microstructures in living tissues is one of the keys to diagnosing early stages of pathology and understanding disease mechanisms. However, the extraction of reliable information on biomarkers based on microstructure details is still a challenge, as the size of features that can be resolved with noninvasive magnetic resonance imaging (MRI) is orders of magnitude larger than the relevant structures. Here we derive from quantum information theory the ultimate precision limits for obtaining such details by MRI probing of water-molecule diffusion. We show that currently available MRI pulse sequences can be optimized to attain the ultimate precision limits by choosing control parameters that are uniquely determined by the expected size, the diffusion coefficient, and the spin relaxation time $T_2$. By attaining the ultimate precision limit per measurement, the number of measurements and the total acquisition time may be drastically reduced compared to the present state of the art. These results are expected to open alternative avenues towards unraveling diagnostic information by quantitative MRI.

Figure 1

Diffusion-weighted spin-echo probing of molecular diffusion. (a) Pulsed gradient spin-echo (PGSE) sequence: an initial $\pi /2$-excitation pulse is followed by the diffusion-weighting contrast block consisting of a $\pi$ rf pulse at half the evolution time $t$ between two-pulsed gradients of duration $\delta$ separated by a delay $\mathrm{\Delta }$. The $\pi$ pulse refocuses external magnetic field inhomogeneities and modulates the effective gradient $G(t)$, switching its sign as shown in the bottom of the scheme. The remaining signal is measured using MRI encoding during the acquisition time (ACQ). (b) Normalized displacement power spectrum $S({\ell }_c,\omega )$ of the restricted molecular diffusion (inset schematic) and a suitable MGSE filter function $F_t(\omega )$ as a function of the renormalized frequency $\omega {\ell }_c^2/2D_0$. The PGSE filter assumes that $\delta =\mathrm{\Delta }=t/2$.

Figure 2

Attaining the ultimate precision bound of the restriction length ${\ell }_c$ from diffusion-weighted magnetization decay. (a) Magnetization decay $M(t)/M(0)$ under Hahn MGSE control as a function of the evolution time in units of the ratio between the diffusion and dephasing lengths ${\ell }_D^2/{\ell }_G^2$. Dark to light green lines denote increasing renormalized correlation times in units of the ratio between the restriction and dephasing lengths ${\ell }_c^2/{\ell }_G^2=({\gamma }^2{G}^2{D}_0{)}^{1/3}{\tau }_c=0.1,0.15,0.25,0.4,1$. The circles denote the optimal evolution times $({\gamma }^2{G}^2{D}_0{)}^{1/3}{t}_{\mathrm{opt}}$. (b) Red circles show the relative minimum error ${\epsilon }^2/{\epsilon }_0^2$ in the estimation of ${\ell }_c$ as a function of ${\ell }_c^2/{\ell }_G^2$ that yields the number of measurements $\mathcal{N}$ needed to attain the ultimate precision bound per measurement. Blue circles show the corresponding value of ${\ell }_D^2/{\ell }_G^2$ as a function of ${\ell }_c^2/{\ell }_G^2$. The bound is approached when ${\ell }_c^2/{\ell }_G^2\ll 1$ and ${\ell }_D^2/{\ell }_G^2\gg 1,$ and $({\ell }_c^4/{\ell }_G^4)({\ell }_D^2/{\ell }_G^2)=-\ln M_0$ as described in Eq. (9), implying that $({\gamma }^2{G}^2{D}_0{)}^{1/3}{\tau }_c\ll 1$.

Figure 3

Optimal control parameters for the estimation of the restriction length ${\ell }_c$. We consider the PGSE sequence with $\mathrm{\Delta }=\delta$ (Hahn) and typical diffusion coefficient $D_0=1\times {10}^{-5}\mathrm{c}{\mathrm{m}}^2/\mathrm{s}$ and relaxation time $T_2=0.1\mathrm{s}$ for the brain’s gray-white matter. (a) Normalized Cramer-Rao error ${\epsilon }_{\mathrm{CR}}^2/{\epsilon }_0^2$ in the estimation of ${\ell }_c$ as a function of time and gradient strength $G$ for a cylindrical compartment with diameter $d=10\mu \mathrm{m}$ (${\ell }_c=0.37d$), typical for axons or microfibers. The top inset shows ${\epsilon }_{\mathrm{CR}}^2/{\epsilon }_0^2$ versus $G$ for an evolution time $t=43$ ms, shown by the white dashed line in the main panel. (b) The normalized relative error ${\epsilon }^2/{\epsilon }_0^2$ as a function of $G$ and diameter $d$. The top inset shows ${\epsilon }^2/{\epsilon }_0^2$ for $d=1,5,10,20\mu \mathrm{m}$. The number of measurements required to attain ${\epsilon }_0$ under the optimal conditions grows as the diameter $d$ increases or $T_2$ decreases. Equivalently, the range of optimal $G$ values that approximate ${\epsilon }_0$ is reduced as ${\tau }_c=\frac{1}{2}{\ell }_c^2/D_0$ approaches $T_2$ (Eq. (13)).